Theorem moan 2114

Description: "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moan	⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))

Proof of Theorem moan

Step	Hyp	Ref	Expression
1		simpr 110	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜑)
2	1	moimi 2110	1 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∃*wmo 2046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by:  moani  2115  mooran1  2117  mormo  2713  rmoan  2964